Question

Evaluate the indefinite integral.

Answer 1

\[\int\limits (x ^{2}+1)(x ^{3}+3x)^{4}dx\]

Answer 2

Substitute \(u=x^3+3x\) and differentiate this with respect to \(x\), try to use this substitution to solve your problem.

Answer 3

ok is du=3x^2 + 3dx and dx=1/3x^2 +1/3

Answer 4

\[\large \frac{du}{dx}=3x^2+3 = 3(x^2+1) \implies \frac{1}{3}\frac{du}{dx}=\underbrace{x^2+1}_! \]

Answer 5

you see how to use this for your integral? \[ \large\int\limits \underbrace{(x ^{2}+1)}_!(x ^{3}+3x)^{4}dx \]

Answer 6

yeah i see

Answer 7

so how would the integral look like, would it look like\[\frac{ 1 }{ 3}\int\limits (u)^{4}du\]

Answer 8

exactly

Answer 9

which is very easy to integrate, just remember the basic rule of thumb "Every substitution requires a back substitution". So evaluate the above integral and then back substitute your \(u\)